Problem: There are 30 students in Mrs. Taylor's kindergarten class.  If there are twice as many students with blond hair as with blue eyes, 6 students with blond hair and blue eyes, and 3 students with neither blond hair nor blue eyes, how many students have blue eyes?
Answer: Let the number of blue-eyed students be $x$, so the number of blond students is $2x$.  Since the number of blue-eyed blond students is $6$, the number of blue-eyed non-blond students is $x - 6$, while the number of blond non-blue-eyed students is $2x - 6$.  Since the number of non-blue-eyed non-blond students is $3$, we can add up these four exclusive categories (blond blue-eyed, blond non-blue-eyed, etc.) to sum to 30 students in the class.

[asy]
unitsize(0.05cm);
label("Blue eyes", (2,74));
label("Blond hair", (80,74));
label("Neither: 3", (44,10));
draw(Circle((30,45), 22));
draw(Circle((58, 45), 22));
label("$6$", (44, 45));
label(scale(0.8)*"$x-6$",(28,58));
label(scale(0.8)*"$2x-6$",(63,58));
[/asy]

So $(x - 6) + (2x - 6) + 6 + 3 = 30$ and $x = \boxed{11}$, which is the number of blue-eyed students.